With the opening of the field of gravitational-wave (GW) astronomy and cosmology [1]}, [2]}, [3]}, we are beginning to use GWs as probes of the Universe. For some cosmological observables, GWs are complementary to electromagnetic observations. This is true in particular for the Hubble parameter \(H_0\) , for which GWs can provide a measurement with different systematics, that can help to eventually resolve the tension between
early-Universe [4]}, [5]}
and late-Universe [6]}, [7]}, [8]} measurements of \(H_0\) . However, there is also cosmological information that can be obtained uniquely through GWs. In general,
on cosmological scales it is convenient to perform a separation between a
homogeneous background described by a Friedmann-Robertson-Walker (FRW) metric, and scalar, vector and tensor perturbations over it. The possibility of deviations from GR in the sector of tensor perturbations can only be investigated using GWs, so this is a genuinely new window that we are beginning to open. In this sector, the most immediate possibility is a deviation of the speed of GWs from the speed of light, but the multi-messenger observation of the binary neutron star (BNS) coalescence
GW170817 [2]}, [10]}, [11]}, [12]} has put an extremely strong limit on it, at the level \(|c_{\rm gw}-c|/c< {\cal O}(10^{-15})\) . However, there is much more than the speed of GWs to be investigated in the tensor sector. In particular, in the last few years it has been appreciated that
all modified gravity models that pass the constraint on the speed of GWs still predict a different evolution of the amplitude of GWs in their propagation over cosmological distances.
This is determined by the equation for tensor perturbations over FRW that, in GR, takes the form
\(\tilde{h}^{\prime \prime }_A +2 {\cal H} \tilde{h}^{\prime }_A+c^2k^2\tilde{h}_A=0\, ,\) 
